From a CFMI technical training course.
“INTRODUCTION TO THE CFM56 FAMILY TRAINING MANUAL
The CFM56 engine, a high by-pass, dual rotor, axial flow turbofan, was designed in the mid 70’s.
It is a product of CFMI, CFM International is a company jointly owned by GENERAL ELECTRIC (GE) of the USA, and SOCIETE NATIONALE D’ETUDE ET DE CONSTRUCTIONDE MOTEURS D’AVIATION (SNECMA) of France.
CFMI, with the full backing of parent companies holding equal shares, has a dual function:
-Overall program management, on behalf of both GE and SNECMA
-Single interface with customers for marketing and product support
Jointly owned company
Uses all GENERAL ELECTRIC AND SNECMA resources…
…With work split 50/50”
http://www.cfm56.com/excel/page06.html
The CFM56 core is based on the GE F101 engine (developed for the B-1 bomber) and employs a single-stage high-pressure turbine to drive a nine-stage compressor. Correspondingly, a Snecma advanced four- or five-stage, low-pressure turbine drives the Snecma fan and booster.
Also the engine gearbox is manufactured by Snecma.
http://www.cfm56.com/excel/page05.html
The resources of GE and Snecma support the production of CFM56 engines. At dozens of sites in North America and France, components are produced, engines are assembled and engine testing is conducted.